The topology in the quantum Hall effect originates from a length scale competition between the magnetic field and the translation invariance of the electron's kinetic energy [1]}, [2]}. When the electrons hop on a lattice, this competition can also lead to incommensurate situations, where the system is no longer invariant to translations [3]}. Such is also the case in quasicrystals, which are ordered but non-periodic solid structures [4]}, [5]}. The long-range order of quasicrystals can be projected from periodic higher-dimensional systems [6]}, [7]}, [8]}, and thus exhibits sharp diffraction peaks [9]}, [10]}, [11]}. Spectral characteristics of such quasiperiodic patterns have been studied in a variety of platforms, revealing highly fractal energy
levels with infinitely many energy gaps, self-similar eigenstates and critical localization [12]}, [13]}, [14]}, [15]}, [16]}, [17]}. Interestingly, the topological characterization of quasicrystals can also be mapped to higher-dimensions using a mapping to the quantum Hall effect, and associating bulk gaps with Chern numbers [18]}, [19]}, [20]}, [21]}, [22]}. This mapping complements earlier topological classifications of quasiperiodic systems by gap labeling approach [23]}, [24]}, [25]}.
